Question

Simplify \[{{2}^{30}}+{{2}^{30}}+{{2}^{30}}+{{2}^{30}}?\]

Answer 1

Hint: To get the solution of this question use the concept of exponents. Take the common term out of all the given terms, after taking common simplify the expression then again express the term in the form of exponents and use the rules of the exponents. Multiply the base with itself by exponent times.

Complete step by step answer:
The given question is in the form of exponents so for solving this type of questions we have to apply the rules of exponents. But what does the term exponent mean? It is simply referred to as the method of performing multiplication various times by the number itself. If we take the example as \[{{12}^{3}}\], here \[12\] is referred to as the base and \[3\] as the power or exponent. And \[{{12}^{3}}\] means that we have to multiply the number \[12\] by the number \[12\] itself \[3\] times. There are also several rules that are used to perform the exponent operation.
To solve this question take \[{{2}^{30}}\] common from all the four terms then we get,
\[\Rightarrow {{2}^{30}}(1+1+1+1)\]
The operator used in this expression after taking common is \[+\] i.e. we have to do summation of \[1\] by \[4\] times. For performing this operation either add \[1\] by \[4\] times or multiply \[1\] by \[4\]by both the operations we will get the same result. You can use any of them.
After simplifying the above expression, we get
\[\Rightarrow {{2}^{30}}\times 4\]
Now express \[4\] in terms of exponent of \[2\] as \[{{2}^{2}}\], we will get
\[\Rightarrow {{2}^{30}}\times {{2}^{2}}\]
By the rule of exponent, the product of powers i.e. when the bases are in product then add the powers together. For example: \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\], here \[a\] is known as the base of the exponent, \[m\] and \[n\] are the powers .
So, after performing this rule of exponent, we will get
\[\Rightarrow {{2}^{32}}\]
Hence we can say that the final answer to the given question is \[{{2}^{32}}=4294967296\].

Note: The everyday example of exponents is if we have to multiply the same number by itself then instead of multiplying that number again and again we just express them in the form of power and exponents. For example: if the length, breadth and height of any room is 12 feet then volume can be expressed as \[{{12}^{3}}\]cubic feet.